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x + y + z divides x^2 + y^2 + z^2 when x^3=y^3=z^3 same remainder mod p

Source: OLCOMA Costa Rica National Olympiad, Final Round, 2016 Shortlist N2 day2

September 24, 2021
dividesdivisiblenumber theory

Problem Statement

Let x,y,zx, y, z be positive integers and pp a prime such that x<y<z<px <y <z <p. Also x3,y3,z3x^3, y^3, z^3 leave the same remainder when divided by pp. Prove that x+y+zx + y + z divides x2+y2+z2x^2 + y^2 + z^2.
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